This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

This paper studies a notion called polynomial-time membership comparable sets. For a function g, a set A is polynomial-time g-membership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)-tt -reducible to a Pselective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` P-mc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...

We take a fresh look at CD complexity, where CD (x) is the size of the smallest program that distinguishes x from all other strings in time t(jxj). We also look at CND complexity, a new nondeterministic variant of CD complexity, and time-bounded Kolmogorov complexity, denoted by C complexity.

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S 2 = S 2 . Building on this, we strengthen the Kamper-- AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap's Theorem, namely, we prove that if NP coNP/poly then the polynomial hierarchy collapses to S 2 . Under the same assumptions, the best previously known collapses were to ZPP respectively ([KW98, BCK 94], building on [KL80, AFK89, Kam91, Yap83]). It is known that S 2 [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kamper-- AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results.

A semi-membership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent half-decade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semi-membership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semi-membership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semi-membership algorithms have been studied in a number of settings. Recursive semi-membership algorithms (and the associated semi-recursive sets---those sets having recursive semi-membership algorithms) were introduced in the 1...

Ko [Ko83] proved that the P-selective sets are in the advice class P/quadratic. We prove that the P-selective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," usually a polynomial-time deterministic or nondeterministic function, that selects which of any two given inputs is logically no less likely than the other to belong to the given set. Definition 1.1 [HNOS94] Let FC be any class of functions (possibly multivalued or partial). A set A is FC-selective if there is a function f 2 FC such that for every x and y, it holds that f(x; y) ` fx; yg, and if fx; yg " A 6= ;, then f(x; y) 6= ; and f(x; y) ` A. Let FC-sel denote the class of sets that are FC-selective. The class that would be notated FP single\Gammavalued; total -sel according to the definition above was defined directly by Selman in 1979 [Sel79]. Henceforward, we refer to these sets ...

. This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that 1. A multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses. Key words. computational complexity, complexity classes, relativized computation, bounded query classes, Boolean hierarchy, multivalued functions, NPMV AMS subject classifications. 68Q05, 68Q10, 68Q15, 03D10, 03D15 1.

Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomial-size circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.

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We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.